Well, it is prime, but if we relax the requirement that it has no integer factors to allow complex numbers with integer components, i.e., a + bi where a and b are integers, then 5 is not prime.
(2 + i)(2 - i) = 4 -2i + 2i -i^2 = 4 - (-1) = 5
I love the BBC 4 program In Our Time. Listen to the discussion of the history of complex numbers at http://downloads.bbc.co.uk/podcasts/radio4/iot/iot_20100923-1035a.mp3
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